HYACINTHOS 24665

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Oa, Ob, Oc = the circumcenters of OBC, OCA, OAB, resp.

N1, N2, N3 = the NPC centers of OObOc, OOcOa, OOaOb, resp

ABC, N1N2N3 are orthologic.
The orthologic center (ABC, N1N2N3) lies on the circumcircle of ABC
The orthologic center (N1N2N3, ABC) lies on the circumcircle of N1N2N3.

*** The orthologic center (ABC, N1N2N3) is X(74)= isogonal conjugate of the point in which the Euler line meets the line at infinity.

*** The orthologic center (N1N2N3, ABC) is  V= 5 X(3) - X(64)

V =( a^2 (2 a^8-5 a^6 (b^2+c^2)+a^4 (3 b^4+4 b^2 c^2+3 c^4)+a^2 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^2 (b^4+c^4)): ... : ...),

with (6-9-13)-search number (0.571122601235493, -0.935307626992647, 4.02459010002445).

V lies on lines: {2, 9833}, {3, 64}, {4, 1495}, {5, 5944}, {6, 3517}, {22, 1092}, {24, 184}, {25, 578}, {26, 206}, {30, 5448}, {39, 1971}, {49, 52}, {51, 54}, {110, 5562}, {125, 10018}, {140, 1503}, {143, 5097}, {156, 1658}, {159, 182}, {161, 569}, {185, 186}, {216, 3463}, {376, 5878}, {394, 9715}, {436, 8884}, {468, 6146}, {549, 6247}, {550, 1511}, {567, 9920}, {568, 9704}, {575, 2393}, {1181, 3515}, {1216, 7502}, {1660, 6644}, {1853, 3526}, {1899, 3147}, {1970, 3199}, {1994, 9706}, {2781, 7555}, {3060, 9545}, {3270, 9638}, {3292, 7556}, {3522, 5656}, {3528, 6225}, {3530, 6696}, {3534, 5895}, {3574, 7576}, {3917, 7512}, {5010, 6285}, {5050, 9924}, {5447, 7525}, {5449, 10020}, {5480, 7715}, {5651, 7509}, {5889, 9544}, {5894, 8703}, {6001, 7508}, {6102, 7575}, {6243, 9703}, {7280, 7355}, {8681, 9937}, {8718, 9934}.

Angel Montesdeoca